Lemma 67.49.3 (0GMC)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.49: Normalization
Lemma 67.49.3 (cite)

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Lemma 67.49.3. Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of algebraic spaces over $S$ . Then for $x \in |X|$ we have: $x$ has codimension $0$ in $X \Rightarrow f(x)$ has codimension $0$ in $Y$ .

Proof. Via Properties of Spaces, Lemma 66.11.1 and étale localization this translates into the case of a morphism of schemes and generic points of irreducible components. Here the result follows as generalizations lift along flat morphisms of schemes, see Morphisms, Lemma 29.25.9. $\square$

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